To find the area $A$ of the region bounded between the curve $f(x) = \ln(x) + 1$ and the line $g(x) = -4x + 6$ over the interval $[1, 4]$, we follow these steps:

Step 1: Set up the area formula

The area between two curves $f(x)$ and $g(x)$ over the interval $[a, b]$ is given by: $A = \int_a^b \left( f(x) - g(x) \right) dx$ Here, $f(x) = \ln(x) + 1$ and $g(x) = -4x + 6$.

Step 2: Write the integral

The area $A$ is: $A = \int_1^4 \left( (\ln(x) + 1) - (-4x + 6) \right) dx$

Simplifying the integrand: [ (\ln(x) + 1) - (-4x + 6) = \ln(x) + 1 +